340 
FROM PLATO TO EUCLID 
of III. 21 that angles in the same segment are equal, a proposi 
tion which Euclid’s more general proof does not need to use. 
These instances are sufficient to show that Euclid was far 
from taking four complete Books out of an earlier text-book 
without change; his changes began at the very beginning, 
and there are probably few, if any, groups of propositions in 
which he did not introduce some improvements of arrange 
ment or method. 
It is unnecessary to go into further detail regarding 
Euclidean theorems found in Aristotle except to note the 
interesting fact that Aristotle already has the principle of 
the method of exhaustion used by Eudoxus: ‘ If I continually 
add to a finite magnitude, I shall exceed every assigned 
(‘ defined ’, coparperov) magnitude, and similarly, if I subtract, 
I shall fall short (of any assigned magnitude).’ 1 
(y) Propositions not found in Euclid. 
Some propositions found in Aristotle but not in Euclid 
should be mentioned. (1) The exterior angles of any polygon 
are together equal to four right angles 2 ; although omitted 
in Euclid and supplied by Prod us, this is evidently a Pytha 
gorean proposition. (2) The locus of a point such that its 
distances from two given points are in a given ratio (not 
being a ratio of equality) is a circle 3 ; this is a proposition 
quoted by Eutocius from Apollonius’s Plane Loci, but the 
proof given by Aristotle differs very little from that of 
Apollonius as reproduced by Eutocius, which shows that the 
proposition was fully known and a standard proof of it was in 
existence before Euclid’s time. (3) Of all closed lines starting 
from a point, returning to it again, and including a given 
area, the circumference of a circle is the shortest 4 ; this shows 
that the study of isoperimetry (comparison of the perimeters 
of different figures having the same area) began long before 
the date of Zenodorus’s treatise quoted by Pappus and Theon 
of Alexandria. (4) Only two solids can fill up space, namely 
the pyramid and the cube 5 ; this is the complement of the 
Pythagorean statement that the only three figures which can 
J Arist. Phys. viii. 10. 266 b 2. 
2 Anal. Post. i. 24. 85 b 38 ; ii. 17. 99 a 19. 
3 Meteorologica, iii. 5. 376 a 3 sq. 4 De caelo, ii, 4. 287 a 27. 
5 lb. iii. 8. '306 b 7.